A-Level Maths: D5-13 Geometric Sequences: 4th term is 24, Sum of the first 4 terms is 21

[nosir]

Would simplifying the a(r⁴–1)/(r–1) to:

a(r²+1)(r²–1)/(r–1)
=
a(r²+1)(r+1)(r–1)/(r–1)
=
a(r²+1)(r+1)

have any drawbacks or potentially lose solutions since I’m dividing through by (r–1)? In this example it worked better than your method since I didn’t get the r=1, so there was no ambiguity in the solutions, but would cancelling in this way be a sustainable method for other examples?

[singularityhq]

How do you decide when to use a(r^n - 1) Vs a(1 - r^n)

[tgc]

I went with
a=312/(1+r+r²+r³) (from u₁+u₂+u₃+u₄=312), a=250/r³
∴ 31r³-125r²-125r-125=0
factor theorem and poly division:
∴ (r-5)(31r²+30r+25)=0 △=-ve, r=5 only real answer